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Several non-standard problems for the stationary Stokes system. (English) Zbl 1437.35586
Summary: This paper studies the Stokes system \(-\Delta{\mathbf{u}}+\nabla\rho={\mathbf{f}}\), \(\nabla\cdot{\mathbf{u}}=\chi\) in \(\Omega\) with three boundary conditions: \[{\mathbf{n}}\cdot{\mathbf{u}}={{\mathbf{n}}\cdot\mathbf{g}}, \quad{\mathbf{n}}\cdot{\mathbf{u}}={\mathbf{n}}\cdot\mathbf{g}, \quad\displaystyle{\mathbf{n}}\cdot{\mathbf{u}}={{\mathbf{n}}\cdot\mathbf{g}}.\] Here \(\Omega\) is a bounded simply connected planar domain. We find a necessary and sufficient condition for the existence of a solution in Sobolev spaces \(W^{s,q}(\Omega;{\mathbb{R}}^2)\times W^{s-1,q}(\Omega)\), with \(1+1/q<s<\infty\), in Besov spaces \(B_s^{q,r}(\Omega;{\mathbb{R}}^2)\times B_{s-1}^{q,r}(\Omega)\), with \(1+1/q<s<\infty\), and classical solutions in \(\mathcal{C}^{k,\alpha}(\overline{\Omega},{\mathbb{R}}^2)\times{\mathcal{C\%}}^{k-1,\alpha}(\overline{\Omega})\), with \(0<\alpha<1\), \(k\in{\mathbb{N}}\).
MSC:
35Q35 PDEs in connection with fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A09 Classical solutions to PDEs
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
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